Solvability of polynomial equations over finite fields

DIMACS Theoretical Computer Science Seminar
Title: Solvability of polynomial equations over finite fields
Speaker: Neeraj Kayal, IAS
Date: Wednesday, October 4, 2006 11:00am - 12:00pm
Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus,
Piscataway, NJ
We investigate the complexity of the following polynomial polynomial solvability
problem - given a finite field
and a set of polynomials
of total degree at most
, determine the existence of
-solution to the system of equations
That is determine if there exists a point
such that
The problem is easily seen to be NP-complete even when the field size is and the
degree of each polynomial is also bounded by . Here we investigate the deterministic
complexity of this problem when the number of variables in the input is bounded. We
show that for any fixed , there is a deterministic algorithm for this problem whose
running time is bounded by a polynomial in
. Moreover the algorithm can
be implemented parallely to get a family of P-uniform circuits of size
and depth
for this problem